Keep in mind, an object does not actually change shape during rotation. Points are colored lighter when closer to the center of the 3D projection (NOT the center of the 4D sphere) just so it’s easier to see where they are in relation to each other. Points that are further away on the w axis will appear closer towards the center of the 3D projection, allowing us to squeeze all of the 4D points into this 3D universe at the same time. This object was constructed by placing random points on the surface of a 4D sphere. Let’s see what we can learn from perspective projection. We haven’t had to worry about rotations yet because rotating a 4D sphere doesn't affect the shape of its slice, but slices don't always give the full picture. At each slice of a 3D sphere an entire 2D universe can be defined, and none of the circular slices intersect each other. The lack of intersection may seem strange, but consider again what happens one dimension lower. This shows that the 3D sphere slices aren't lined up on any of the 3 axes we're familiar with, and also that none of them are intersecting. The way I like to think about it is that an entire separate 3D universe exists for each slice. Realize that this is the equivalent of a 2D being trying to form a sphere by lining up circle slices like this: ⚠️ WARNING ⚠️ If you find yourself trying to line up these slices in your head – something like this: It's just like our 3D sphere that passed through a 2D universe, creating circular slices whose size changed with z distance! When the 4D sphere passes through our 3D universe, the slices it creates are 3D spheres whose size change with w distance.
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